SIMILARITY SOLUTION FOR UNSTEADY HEAT AND MASS TRANSFER FROM A STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM WITH SUCTION/INJECTION AND CHEMICAL REACTION EFFECTS

An analysis is presented to investigate the effects of chemical reaction on unsteady free convective heat and mass transfer on a stretching surface in a porous medium. The governing partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically using an efficient tri-diagonal implicit finite-difference method. The results obtained show that the flow field is influenced appreciably by the presence of unsteadiness parameter, chemical reaction parameter, permeability parameter, and suction/injection parameter.     

UNSTEADY HEAT AND MASS TRANSFER BY MHD MIXED CONVECTION FLOW OVER AN IMPULSIVELY STRETCHED VERTICAL SURFACE WITH CHEMICAL REACTION AND SORET AND DUFOUR EFFECTS

This work considers unsteady, laminar, and coupled heat and mass transfer by MHD mixed convective boundary-layer flow of an electrically conducting fluid over an impulsively stretched vertical surface in an unbounded quiescent fluid with aiding external flow in the presence of a transverse magnetic field, homogeneous chemical reaction, and Soret and Dufour effects. The stretching velocity and surface temperature and concentration are assumed to vary linearly with the distance along the surface. The flow is impulsively set into motion and both the temperature and concentration at the surface are also suddenly changed from those of the ambient fluid. The governing partial differential equations are transformed into a set of nonsimilar equations and solved numerically by an efficient implicit, iterative, finite-difference method. A parametric study illustrating the influence of various physical parameters is performed. Numerical results for the steady-state velocity, temperature, and concentration profiles as well as the time histories of the skin-friction coefficient, local Nusselt number, and local Sherwood number are presented graphically and discussed.     

UNSTEADY MHD BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A PERMEABLE STRETCHING SHEET WITH SUCTION OR BLOWING

In this study, unsteady MHD boundary layer flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet with suction or blowing and also with power-law variation in wall concentration is investigated. Using similarity transformation, the governing partial differential equations are converted into nonlinear self-similar ordinary differential equations. The transformed equations are then solved by the finite difference method using the quasi-linearization technique. Due to the increase in the unsteadiness parameter, the velocity initially decreases, but after a certain point it increases. A similar effect is also observed in case of concentration distribution. The increase in magnetic parameter causes a decrease in velocity and an increase in concentration. For increasing strength of applied suction both momentum and concentration boundary layer thicknesses decrease. On the other hand, applied blowing has reverse effects. Moreover, the mass transfer from the sheet is enhanced with increasing values of Schmidt number, reaction rate parameter, and also power-law exponent (related to wall concentration distribution). For high negative values of the power-law exponent, mass absorption at the sheet occurs. Moreover, due to increase of unsteadiness, this mass absorption is prevented.     

MASS TRANSFER WITH HETEROGENEOUS CHEMICAL REACTION OF THE FIRST ORDER IN FALKNER-SKAN FLOW OF A POWER-LAW FLUID

Mass transfer with a heterogeneous chemical reaction of the first order in a Falkner-Skan flow of non-Newtonian power-law fluid is investigated. This problem is solved by the method of Laplace transform following the approach suggested by Apelblat (1980). The solution is obtained in a closed analytical form.     

MASS TRANSFER WITH HETEROGENEOUS CHEMICAL REACTION OF THE FIRST ORDER IN FALKNER-SKAN FLOW OF A POWER-LAW FLUID

ABSTRACT

Mass transfer with a heterogeneous chemical reaction of the first order in a Falkner-Skan flow of non-Newtonian power-law fluid is investigated. This problem is solved by the method of Laplace transform following the approach suggested by Apelblat (1980 Apelblat , A. ( 1980 ). Chem. Eng. J. , 19 , 19 – 37 . [CSA] [CROSSREF]  [Google Scholar]). The solution is obtained in a closed analytical form.     

HEAT AND MASS TRANSFER IN STAGNATION-POINT FLOW OF A POLAR FLUID TOWARDS A STRETCHING SURFACE IN POROUS MEDIA IN THE PRESENCE OF SORET,DUFOUR AND CHEMICAL REACTION EFFECTS

This work is focused on the numerical solution of steady boundary-layer stagnation-point flow of a polar fluid towards a stretching surface embedded in porous media in the presence of the effects of Soret and Dufour numbers and first-order homogeneous chemical reaction. The governing boundary-layer equations of the problem are formulated and transformed into a self-similar form. The obtained equations are solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. Both assisting and opposing flow conditions are considered. Comparisons of the present numerical results with previously published work under limiting cases are performed and found to be in excellent agreement. Representative results for the fluid velocity, angular velocity, temperature, and solute concentration profiles as well as the local heat and mass transfer rates for various values of the physical parameters are displayed in both graphical and tabular forms.     

MASS TRANSFER TO A DROPLET WITH A FIRST-ORDER CHEMICAL REACTION

The steady-state mass transfer to a droplet with a first-order chemical reaction is i nvestigated as a function of the Damkohler number (Da₁₁) for the special case of very high Peclet number flow. The flow field in the droplet employed in this work is Hill's spherical vortex. The resistance to mass transfer is assumed to occur inside the droplet with negligible resistance to mass transfer in the continuous phase.     

MASS TRANSFER WITH CHEMICAL REACTION IN PARTIALLY WETTED FLAT PLATE CATALYST PELLETS: RIVULET FLOW ANALYSIS

Mass transfer with chemical reaction is analyzed in a system formed by a flat plate solid catalyst, partially wetted by a flowing rivulet of a liquid in contact with a stagnant pure gas. The paper solves the fluid dynamic problem of the liquid phase first, and afterwards incorporates the mass transfer and the chemical reaction. The system is assumed to be isothermal and at steady state, with a first order kinetics whose limiting reactant is in the gas phase. This work studies the influence of the gas-liquid surface tension, the liquid reactant flow rate, the liquid viscosity and the angle of inclination of the solid, upon the wetting factor. The model proposed also predicts the effect of these parameters and the Thiele modulus on the overall effectiveness factor and the molar flux of the limiting gaseous reactant at the catalytic solid-liquid interface in a direct way. This approach makes the wetting factor a non-manipulated variable.     

SLIP FLOW AND HEAT TRANSFER OF A SECOND-GRADE FLUID IN A POROUS MEDIUM OVER A STRETCHING SHEET WITH POWER-LAW SURFACE TEMPERATURE OR HEAT FLUX

This article deals with the study of boundary layer flow of a second-grade fluid in a porous medium past a stretching sheet and heat transfer characteristics with power-law surface temperature or heat flux. The flow in the boundary layer is considered to be generated solely by the linear stretching of the boundary sheet adjacent to a porous medium, and boundary wall slip condition is assumed. In the energy equation effects of viscous dissipation, work done due to deformation and internal heat generation/absorption is taken into account. Closed form solutions are obtained for this problem.     

A GENERALIZED PERTURBATION APPROACH TO THE ANALYSIS OF MASS TRANSFER WITH CHEMICAL REACTION

A generalized perturbation approach is presented for the analysis of the simultaneous absorption of two gases in a liquid accompanied by a chemical reaction. The main objective is to derive approximate analytical solutions for the enhancement factors of the solute gases, for all reaction rates. The method consists of constructing a regular perturbation solution for slow reaction, whose convergence is accelerated by means of an Euler transformation, thereby extending its range of utility for larger reaction rates. Convergence is further improved by suppressing the leading singularity of the Euler-transformed series. The accelerated solution is found to predict the enhancement factors very accurately in the slow, intermediate and fast reaction regimes, and in particular, approach the boundary layer limit associated with fast reaction.     

MHD FLOW AND HEAT TRANSFER IN A VISCOUS FLUID OVER A NON-ISOTHERMAL STRETCHING SURFACE WITH THERMAL RADIATION IN SLIP-FLOW REGIME

The present work is concerned with the effects of surface slip conditions and thermal radiation on an electrically conducting fluid over a non-isothermal stretching surface in the presence of a uniform transverse magnetic field. Similarity transformation is used to transform the partial differential equations describing the problem into a system of nonlinear ordinary differential equations, which is solved analytically. The effects of various parameters on the velocity and temperature profiles as well as on the local skin-friction and the local Nusselt number are discussed in detail and displayed through graphs.     

HEAT TRANSFER ENHANCEMENT WITH CHEMICAL REACTION-MODEL COMPARISONS

A one dimensional tubular reactor model which incorporates heat transfer enhancement calculations in the presence of homogenous chemical reactions is presented. The model compensates for the distortion in the actual radial temperature profile due to chemical reaction by calculating the temperature profile in the film next to the reactor wall which is then used to correct for the heat transfer coefficient based on the non-reactive case. The relative simplicity and rapidity of the model makes it a viable alternative to the two dimensional model     

FLOW AND HEAT TRANSFER FOR THREE-DIMENSIONAL FLOW OVER AN EXPONENTIALLY STRETCHING SURFACE

This study analyzes the laminar boundary-layer flow and heat transfer characteristics of a steady, three-dimensional viscous fluid driven by a horizontal surface stretched exponentially in two lateral directions. The simulations in this study assume that the surface temperature is also distributed exponentially and reduce the governing equations to a set of ordinary differential equations using a similarity transformation. This study develops a numerical procedure that combines the Ackroyd method and Runge-Kutta integration scheme to solve the transformed equations. Results show that heat transfer characteristics depend strongly on the stretching ratio, temperature exponent, and Prandtl number.     

UNSTEADY MHD HEAT AND MASS TRANSFER BY MIXED CONVECTION FLOW IN THE FORWARD STAGNATION REGION OF A ROTATING SPHERE AT DIFFERENT WALL CONDITIONS

This study is focused on the problem of MHD heat and mass transfer by mixed convection flow in the forward stagnation region of a rotating sphere in the presence of heat generation and chemical reaction effects. The surface of the sphere is maintained at constant fluid temperature and species concentration. The governing equations of the problem are converted into ordinary differential equations by using suitable similarity transformations. Two cases are considered, namely, constant wall temperature and mass (CWTM) and constant heat and mass fluxes (CHMF). The obtained self-similar equations for both cases are solved numerically using an efficient iterative implicit finite-difference method. The numerical results are compared with previously published results on special cases of the problem and found to be in excellent agreement. The obtained results are displayed graphically to illustrate the influence of the different physical parameters on the velocity components in x- and y-directions, temperature, and concentration profiles as well as the local surface shear stresses and local heat and mass transfer coefficients.     

HEAT TRANSFER DUE TO MHD SLIP FLOW OF A SECOND-GRADE LIQUID OVER A STRETCHING SHEET THROUGH A POROUS MEDIUM WITH NONUNIFORM HEAT SOURCE/SINK

An analysis is carried out to study the heat transfer characteristics of a second-grade non-Newtonian liquid due to a stretching sheet through a porous medium under the influence of external magnetic field. The stretching sheet is assumed to be impermeable. Partial slip condition is used to study the flow behavior of the liquid. The effects of viscous dissipation, nonuniform heat source/sink on the heat transfer are addressed. The nonlinear partial differential equations governing momentum and heat transfer in the boundary layer are converted into nonlinear ordinary differential equations using similarity transformation. Analytical solutions are obtained for the resulting boundary value problems in the case of two types of boundary heating, namely, constant surface temperature (CST) and prescribed surface temperature (PST). The effects of slip parameter, second-grade liquid parameter, combined (magnetic and porous) parameter, Prandtl number, Eckert number, and nonuniform heat source/sink parameters on the heat transfer are shown in several plots. Analytical expressions for the wall frictional drag coefficient and wall temperature gradient are obtained.     

HYDROMAGNETIC STAGNATION POINT FLOW TOWARDS A POROUS STRETCHING SHEET WITH VARIABLE SURFACE HEAT FLUX IN THE PRESENCE OF HEAT GENERATION

The influence of heat generation or absorption on the steady, two-dimensional flow of an electrically conducting fluid near a stagnation point on a stretching permeable surface with variable surface heat flux in the presence of a magnetic field is investigated. The governing system of partial differential equations describing the problem are converted into highly non-linear ordinary differential equations using similarity transformation. Numerical solutions of these equations are obtained using the fourth-order Runge-Kutta integration scheme with the shooting method. The effects of the heat generation or absorption parameter and the velocity ratio parameter on the velocity and the temperature are displayed graphically and discussed. The numerical values of the local skin-friction coefficient and the local Nusselt number for various values of physical parameters are presented through tables and discussed.     

SOME UNSTEADY HEAT TRANSFER PROBLEMS SOLVED BY A SIMPLE SIMILARITY TRANSFORMATION

A similarity transformation is used for solving equations of the form

This equation is reduced by means of the similarity variable to an equation with partial derivatives of the first order for 62 and to an ordinary differential equation of the second order for T. For heat transfer from solid bodies, the preceding equation is obtained when the velocity distribution can be approximated by the potential flow. Such circumstances do occur for liquid metals, because for low Prandtl numbers the thickness of the thermal boundary layer is much larger than that of the hydrodynamic boundary layer. Several cases are examined: (a) Stagnation in planar flow; (b) Flow along a planar plate in unsteady hydrodynamic conditions; (c) Flow along a curved surface in steady conditions. The time in which the steady state is achieved is evaluated.     

MASS TRANSFER WITH CHEMICAL REACTION IN A FROTH BED REACTOR

A model to predict conversions in a froth bed reactor has been developed. The model is then compared against the available experimental data on the oxidation of sodium sulfide in a foam bed contactor. The predictions using the present model are also compared against those based on the model of a foam bed reactor developed earlier. The predictions using the present model agree fairly well with the experimental data and, in some cases, are even in better quantitative agreement than the previous single stage model of a foam bed reactor. The case of significant surface resistance due to surfactant has also been analyzed theoretically, obtaining analytical solutions for the concentration profile and fractional gas absorption in a liquid froth shell.     

MASS TRANSFER OVER AN EXPONENTIALLY STRETCHING POROUS SHEET EMBEDDED IN A STRATIFIED MEDIUM

The boundary layer flow and mass transfer towards an exponentially stretching porous sheet embedded in a stratified medium is presented in this analysis. A first-order constructive/destructive chemical reaction is also considered. Similarity transformations were used to convert the partial differential equations corresponding to the momentum and concentration into highly nonlinear ordinary differential equations. Numerical solutions of these equations were obtained by the shooting method. Mass absorption at the surface was found in the case of a stratified medium, and it increased with an increase of stratification parameter. Due to increasing reaction rate parameter the concentration decreased. It is important to note that concentration overshoot was observed in the case of a stratified medium.     

NUMERICAL MODELING OF MHD CONVECTIVE HEAT AND MASS TRANSFER IN PRESENCE OF FIRST-ORDER CHEMICAL REACTION AND THERMAL RADIATION

An analysis was carried out numerically to study unsteady heat and mass transfer by free convection flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate under the action of transverse magnetic field taking into account thermal radiation as well as homogeneous chemical reaction of first order. The fluid considered here is an optically thin gray gas, absorbing-emitting radiation, but a non-scattering medium. The porous plate was subjected to a constant suction velocity with variable surface temperature and concentration. The dimensionless governing coupled, nonlinear boundary layer partial differential equations were solved by an efficient, accurate, extensively validated, and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields were studied for the effects of Hartmann number (M), radiation parameter (R), chemical reaction (K), and Schmidt number (Sc). The local skin friction, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is found that velocity is reduced considerably with a rise in the magnetic body parameter (M), whereas the temperature and concentration are found to be markedly boosted with an increase in the magnetic body parameter (M). An increase in the conduction-radiation parameter (R) is found to escalate the local skin friction (τ), Nusselt number, and concentration, whereas an increase in the conduction-radiation parameter (R) is shown to exert the opposite effect on either velocity or temperature field. Similarly, the local skin friction and the Sherwood number are both considerably increased with an increase in the chemical reaction parameter. Possible applications of the present study include laminar magneto-aerodynamics, materials processing, and MHD propulsion thermo-fluid dynamics.